Aircraft: B737
Phase of flight: Initial Approach; Climb
Anomaly: Aircraft Equipment Problem Less Severe; Deviation / Discrepancy - Procedural Published Material / Policy

Departing; we got a Leading Edge Transit light after flap retraction. We ran the QRH; contacted Dispatch; and advised them of the situation. After running the QRH; noting the speed and altitude restrictions; the light extinguished. With the concurrence of Dispatch; we decided we could go normal speed and continue to our destination; since there was no longer an indication of an abnormality. The rest of the cruise portion the flight was uneventful.As we made our initial descent; I began to have audio control panel issues. I was not able to hear the Captain over the intercom; and couldn't hear ATC on Comm 1 either. We turned up the speaker and communicated with each other without the use of intercom for the duration of the flight; although I was able to hear Comm 1 again after passing through about 4;000 feet AGL. However; communication via the intercom wasn't regained until reaching the ground. ATC told us to expect a visual approach. We were cleared for the visual and switched to Tower when they cleared us to switch to another runway and then cleared us to land. We went flaps 30; slowed to target speed (133 knots). At 1;400 feet AGL and 133 knots; we got the stick shaker and Airspeed Low aural warning. We did a go around; and in the process of cleaning up; we got the Leading Edge Transit light once again.On the go-around; the flight director and autopilot weren't available. Eventually on the base to final on our second approach; I regained the flight director; but didn't regain use of the autopilot. We again ran the QRH for Leading Edge Transit light and got vectors. ATC then changed the runway again as we were on base to final. While we were running the QRH; my FMS locked up while attempting to change the runway. Since it was visual; we just decided to acquire the runway visually and back it up quickly with the Localizer frequency. While on final; we configured for flaps 15; landing per the QRH; and discussed the possibility of again getting the stick shaker and that we would maintain target for VREF 15 + 15 (151 knots) until touchdown and continue the approach in the event we get another erroneous stick shaker. At 1;500 feet we again got a momentary stick shaker and continued the approach; as briefed. We also missed the Flap Inhibit switch on the QRH Checklist and got a Too Low Flaps warning. Realizing that this was a result of the Flap Inhibit switch not being switched; the Captain immediately selected Flap Inhibit and we continued the approach and landed. We were met by Maintenance at the gate and put the numerous issues (autopilot; ACP; Leading Edge light; erroneous stick shaker; FMS lock up) in the book.With the benefit of hindsight; there were many additive factors that led this to being a stressful situation. The task loading was significant for both Pilots; and I believe we could have brought ATC into the loop; which would have eased the task loading. The perception that we were being rushed and the communication difficulties led to us missing the Flap Inhibit switch. Air crew experienced several minor mechanical issues with the biggest being an erroneous stick shaker activation during configuration to landing. Crew overcame the issues and landed normally.